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Number of chains of distinct strictly superior divisors starting with n.
24

%I #21 Nov 02 2024 09:13:56

%S 1,1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,9,1,2,2,4,1,7,1,6,2,2,

%T 2,10,1,2,2,9,1,6,1,4,4,2,1,19,1,4,2,4,1,8,2,9,2,2,1,20,1,2,4,10,2,6,

%U 1,4,2,7,1,29,1,2,4,4,2,6,1,19,3,2,1,19,2

%N Number of chains of distinct strictly superior divisors starting with n.

%C We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

%C These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).

%C Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

%H Amiram Eldar, <a href="/A342084/b342084.txt">Table of n, a(n) for n = 1..10000</a>

%F a(2^n) = A045690(n).

%e The a(n) chains for n = 2, 6, 12, 16, 24, 30, 32, 36:

%e 2 6 12 16 24 30 32 36

%e 6/3 12/4 16/8 24/6 30/6 32/8 36/9

%e 12/6 16/8/4 24/8 30/10 32/16 36/12

%e 12/6/3 24/12 30/15 32/8/4 36/18

%e 24/6/3 30/6/3 32/16/8 36/12/4

%e 24/8/4 30/10/5 32/16/8/4 36/12/6

%e 24/12/4 30/15/5 36/18/6

%e 24/12/6 36/18/9

%e 24/12/6/3 36/12/6/3

%e 36/18/6/3

%e The a(n) ordered factorizations for n = 2, 6, 12, 16, 24, 30, 32, 36:

%e 2 6 12 16 24 30 32 36

%e 3*2 4*3 8*2 6*4 6*5 8*4 9*4

%e 6*2 4*2*2 8*3 10*3 16*2 12*3

%e 3*2*2 12*2 15*2 4*2*4 18*2

%e 3*2*4 3*2*5 8*2*2 4*3*3

%e 4*2*3 5*2*3 4*2*2*2 6*2*3

%e 4*3*2 5*3*2 6*3*2

%e 6*2*2 9*2*2

%e 3*2*2*2 3*2*2*3

%e 3*2*3*2

%t ceo[n_]:=Prepend[Prepend[#,n]&/@Join@@ceo/@Select[Most[Divisors[n]],#>n/#&],{n}];

%t Table[Length[ceo[n]],{n,100}]

%Y The restriction to powers of 2 is A045690, with reciprocal version A040039.

%Y The inferior version is A337135.

%Y The strictly inferior version is A342083.

%Y The superior version is A342085.

%Y The additive version allowing equality is A342094 or A342095.

%Y The additive version is A342096 or A342097.

%Y A000005 counts divisors.

%Y A001055 counts factorizations.

%Y A003238 counts divisibility chains summing to n-1, with strict case A122651.

%Y A038548 counts inferior (or superior) divisors.

%Y A056924 counts strictly inferior (or strictly superior) divisors.

%Y A067824 counts strict chains of divisors starting with n.

%Y A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A207375 lists central divisors.

%Y A253249 counts strict chains of divisors.

%Y A334996 counts ordered factorizations by product and length.

%Y A334997 counts chains of divisors of n by length.

%Y - Inferior: A033676, A063962, A066839, A072499, A161906.

%Y - Superior: A033677, A070038, A161908, A341591.

%Y - Strictly Inferior: A060775, A070039, A333806, A341674.

%Y - Strictly Superior: A064052/A048098, A140271, A238535, A341642, A341673.

%Y Cf. A000203, A000929, A001248, A006530, A018819, A020639, A169594, A337105, A342098.

%K nonn

%O 1,6

%A _Gus Wiseman_, Feb 28 2021